L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 4·11-s + 13-s + 4·14-s + 16-s + 4·17-s + 7·19-s + 4·22-s − 4·23-s − 26-s − 4·28-s − 5·29-s + 4·31-s − 32-s − 4·34-s + 9·37-s − 7·38-s + 5·41-s − 10·43-s − 4·44-s + 4·46-s − 3·47-s + 9·49-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s + 0.852·22-s − 0.834·23-s − 0.196·26-s − 0.755·28-s − 0.928·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s + 1.47·37-s − 1.13·38-s + 0.780·41-s − 1.52·43-s − 0.603·44-s + 0.589·46-s − 0.437·47-s + 9/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84839466627882260531660648221, −7.17820801860370475579712412526, −6.36131625563931173652001453945, −5.76401865220602795492935628012, −5.07341508593951813406554916569, −3.72568082704772565766838040552, −3.14004679816178465453479694250, −2.42523891159489288106708860431, −1.07539534286625485909191197391, 0,
1.07539534286625485909191197391, 2.42523891159489288106708860431, 3.14004679816178465453479694250, 3.72568082704772565766838040552, 5.07341508593951813406554916569, 5.76401865220602795492935628012, 6.36131625563931173652001453945, 7.17820801860370475579712412526, 7.84839466627882260531660648221